| L(s) = 1 | + 1.49·3-s + 5-s + 4.58·7-s − 0.777·9-s − 2.58·11-s + 2.75·13-s + 1.49·15-s + 17-s − 2.07·19-s + 6.83·21-s + 4.58·23-s + 25-s − 5.63·27-s + 4.07·29-s + 0.509·31-s − 3.85·33-s + 4.58·35-s − 0.981·37-s + 4.11·39-s + 9.13·41-s + 8.53·43-s − 0.777·45-s − 6.61·47-s + 14.0·49-s + 1.49·51-s − 13.2·53-s − 2.58·55-s + ⋯ |
| L(s) = 1 | + 0.860·3-s + 0.447·5-s + 1.73·7-s − 0.259·9-s − 0.779·11-s + 0.765·13-s + 0.384·15-s + 0.242·17-s − 0.476·19-s + 1.49·21-s + 0.956·23-s + 0.200·25-s − 1.08·27-s + 0.756·29-s + 0.0914·31-s − 0.670·33-s + 0.775·35-s − 0.161·37-s + 0.658·39-s + 1.42·41-s + 1.30·43-s − 0.115·45-s − 0.964·47-s + 2.00·49-s + 0.208·51-s − 1.81·53-s − 0.348·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.258893502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.258893502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 1.49T + 3T^{2} \) |
| 7 | \( 1 - 4.58T + 7T^{2} \) |
| 11 | \( 1 + 2.58T + 11T^{2} \) |
| 13 | \( 1 - 2.75T + 13T^{2} \) |
| 19 | \( 1 + 2.07T + 19T^{2} \) |
| 23 | \( 1 - 4.58T + 23T^{2} \) |
| 29 | \( 1 - 4.07T + 29T^{2} \) |
| 31 | \( 1 - 0.509T + 31T^{2} \) |
| 37 | \( 1 + 0.981T + 37T^{2} \) |
| 41 | \( 1 - 9.13T + 41T^{2} \) |
| 43 | \( 1 - 8.53T + 43T^{2} \) |
| 47 | \( 1 + 6.61T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 6.61T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 2.11T + 73T^{2} \) |
| 79 | \( 1 + 2.01T + 79T^{2} \) |
| 83 | \( 1 - 7.01T + 83T^{2} \) |
| 89 | \( 1 + 6.33T + 89T^{2} \) |
| 97 | \( 1 - 8.65T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.760137704341752767963698271608, −7.985914824462653568574377455507, −7.79142057574260340486822931482, −6.55808384766428851501818416198, −5.58405510421466323137112412270, −4.96391112156628405822200420022, −4.07030259168943969517793733414, −2.92034984397481744321663336312, −2.19609042467889955519622299188, −1.18933520889511397129741156969,
1.18933520889511397129741156969, 2.19609042467889955519622299188, 2.92034984397481744321663336312, 4.07030259168943969517793733414, 4.96391112156628405822200420022, 5.58405510421466323137112412270, 6.55808384766428851501818416198, 7.79142057574260340486822931482, 7.985914824462653568574377455507, 8.760137704341752767963698271608
